Subgroups And Structure Of Finite Groups,?-supersolvable And Semi-?-nilpotent Groups,Subgroup Lattices And ?-local Formations

This PHD thesis mainly studies the influence of some properties of subgroups on the structure of finite groups and subgroup lattice,establishes the theories of ?-supersolvable groups and semi-?-nilpotent groups,and the theories of ?-local and n-multiply a-local formations.This thesis is divided into six chapters.In chapter ?,we introduce the research background and main results of this disser?tation.In chapter ?,we give some notations,terminologies and some known basic results.In chapter ?,we study some properties of subgroups and the structure of finite groups.In the first section,we study the prime spectrum of maximal subgroups of fi-nite groups.By using some well-known number theoretical results,we solved an open problem on the prime spectrum of maximal subgroups proposed by Monakhov and A.N.Skiba.In the second section,we study the influence of weakly ?-permutable sub-groups on the structure of finite groups.Combining the two definitions of ?-permutable subgroups and weakly s-permutable subgroups proposed by A.N.Skiba,we propose a new definition of weakly a-permutable subgroups.By studying the property of weakly?-permutable subgroups of maximal subgroups of the Hall subgroup of G,we obtain a new discriminant criterion of supersolvability and a normal subgroup of G is hyper-cyclically embedded in G.The results generalize many previous results.In the third section,we study some properties of ?-quasi-F-groups.We give the conditions for a group G to be a ?-quasi-F-group,which give the answer to an open problem about?-quasi-F-groups.In chapter IV,we establish two new classes of finite groups.In the first section,we mainly use the properties of ?-groups proposed by Professor A.N.Skiba and Professor Guo Wenbin to establish the theory of ?-supersolvable groups,and give its detailed characterizations.In the second section,we give the theory of semi-?-nilpotent groups and give the detailed structure of this class of groups.In chapter V,we study two subgroup lattices of finite groups:LCF(G)and LF(G),and give the conditions under which the lattices LF(G)and LCS(G)are coincide.As an application,we prove that a finite soluble group G is a PST-group if and only if LH(G)=LCF(G),where F is the class of all nilpotent groups.In chapter VI,we build the theories of ?-local formations and n-multiply ?-local formations.In the first section,we establish the theory of ?-local formations and gen-eralize the Kramer's theory.In the second section,we generalized the definition of ?-local formations and establish the theory of n-multiply ?-local formations.Moreover,we give some properties and the lattice structure of n-multiply ?-local formations.